Question: Determine how many solutions exist for the system of equations. ${-3x-3y = -15}$ ${x+y = -10}$
Answer: Convert both equations to slope-intercept form: ${-3x-3y = -15}$ $-3x{+3x} - 3y = -15{+3x}$ $-3y = -15+3x$ $y = 5-x$ ${y = -x+5}$ ${x+y = -10}$ $x{-x} + y = -10{-x}$ $y = -10-x$ ${y = -x-10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -x+5}$ ${y = -x-10}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.